Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Harmonic analysis and the geometry of fractals
Izabella Laba
International Congress of Mathematicians
Seoul, August 2014
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Harmonic analysis has long studied singular and oscillatory integrals
associated with surface measures on lower-dimensional manifolds
in Rd . The behaviour of such integrals depends on the geometric
properties of the manifold: dimension, smoothness, curvature.
This is a well established, thriving and productive research area.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
What about the case d = 1? There are no non-trivial
lower-dimensional submanifolds on the line. However, there are
many fractal sets of dimension between 0 and 1. Can we extend
the higher-dimensional theory to this case?
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
What about the case d = 1? There are no non-trivial
lower-dimensional submanifolds on the line. However, there are
many fractal sets of dimension between 0 and 1. Can we extend
the higher-dimensional theory to this case?
If so, what is the right substitute for curvature?
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
What about the case d = 1? There are no non-trivial
lower-dimensional submanifolds on the line. However, there are
many fractal sets of dimension between 0 and 1. Can we extend
the higher-dimensional theory to this case?
If so, what is the right substitute for curvature?
Partial answer: “pseudorandomness,” suggested by additive
combinatorics.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
In additive combinatorics, “pseudorandomness” refers to lack of
additive structure. (The precise formulation depends on the
problem at hand.) This is a key ingredient of major advances on
Szemerédi-type problems (Gowers, Green-Tao), and we will draw
on ideas from that work.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
In additive combinatorics, “pseudorandomness” refers to lack of
additive structure. (The precise formulation depends on the
problem at hand.) This is a key ingredient of major advances on
Szemerédi-type problems (Gowers, Green-Tao), and we will draw
on ideas from that work.
For us, “random” fractals will behave like curved hypersurfaces
such as spheres, whereas structured fractals (e.g. the middle-third
Cantor set) behave like flat surfaces.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Outline of talk:
I
Set-up: measures, dimensionality, curvature/randomness and
Fourier decay.
I
Restriction estimates.
I
Maximal estimates and differentiation theorems.
I
Szemerédi-type results.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Dimensionality of measures
Let µ be a finite, nonnegative Borel measure on Rd .
I
Let 0 ≤ α ≤ d. We say that µ obeys the α-dimensional ball
condition if
µ(B(x, r )) ≤ Cr α ∀x ∈ Rd , r ∈ (0, ∞)
B(x, r ) ball of radius r centered at x.
Izabella Laba
Harmonic analysis and the geometry of fractals
(1)
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Dimensionality of measures
Let µ be a finite, nonnegative Borel measure on Rd .
I
Let 0 ≤ α ≤ d. We say that µ obeys the α-dimensional ball
condition if
µ(B(x, r )) ≤ Cr α ∀x ∈ Rd , r ∈ (0, ∞)
(1)
B(x, r ) ball of radius r centered at x.
I
Connection to Hausdorff dimension via Frostman’s Lemma: if
E ⊂ Rd closed, then
dimH (E ) = sup{α ∈ [0, d] : E supports a probability measure
µ = µα obeying (1)}
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Examples
I
The surface measure σ on the sphere S d−1 obeys the ball
condition with α = d − 1.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Examples
I
The surface measure σ on the sphere S d−1 obeys the ball
condition with α = d − 1.
I
The surface measure on a smooth k-dimensional submanifold
of Rd obeys the ball condition with α = k.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Examples
I
The surface measure σ on the sphere S d−1 obeys the ball
condition with α = d − 1.
I
The surface measure on a smooth k-dimensional submanifold
of Rd obeys the ball condition with α = k.
I
Let E be the middle-third Cantor set on the line, then the
natural self-similar measure on E obeys the ball condition with
2
α = log
log 3 .
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
More general Cantor measures
Construct µ supported on E =
I
T∞
j=1 Ej
via Cantor iteration:
Divide [0, 1] into N intervals of equal length, choose t of
them. This is E1 .
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
More general Cantor measures
Construct µ supported on E =
T∞
j=1 Ej
via Cantor iteration:
I
Divide [0, 1] into N intervals of equal length, choose t of
them. This is E1 .
I
Suppose Ej has been constructed as a union of t j intervals of
length N −j . For each such interval, subdivide it into N
subintervals of length N −j−1 , then choose t of them, for a
total of t j+1 subintervals.This is Ej+1 .
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
More general Cantor measures
Construct µ supported on E =
T∞
j=1 Ej
via Cantor iteration:
I
Divide [0, 1] into N intervals of equal length, choose t of
them. This is E1 .
I
Suppose Ej has been constructed as a union of t j intervals of
length N −j . For each such interval, subdivide it into N
subintervals of length N −j−1 , then choose t of them, for a
total of t j+1 subintervals.This is Ej+1 .
I
The choices of subintervals might or might not be the same at
all stages of the construction, or for all intervals of Ej .
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
More general Cantor measures
Construct µ supported on E =
T∞
j=1 Ej
via Cantor iteration:
I
Divide [0, 1] into N intervals of equal length, choose t of
them. This is E1 .
I
Suppose Ej has been constructed as a union of t j intervals of
length N −j . For each such interval, subdivide it into N
subintervals of length N −j−1 , then choose t of them, for a
total of t j+1 subintervals.This is Ej+1 .
I
The choices of subintervals might or might not be the same at
all stages of the construction, or for all intervals of Ej .
I
Let µj = |E1j | 1Ej , then µj converge weakly to µ, a probability
measure on E .
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
More general Cantor measures, cont.
For any choice of subintervals in the Cantor construction, E has
log t
α
Hausdorff dimension α = log
N , and µ(B(x, r )) ≤ Cr for all
x ∈ R, r > 0.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
More general Cantor measures, cont.
For any choice of subintervals in the Cantor construction, E has
log t
α
Hausdorff dimension α = log
N , and µ(B(x, r )) ≤ Cr for all
x ∈ R, r > 0.
The Fourier-analytic properties of µ depend on the choice of
subintervals.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
More general Cantor measures, cont.
For any choice of subintervals in the Cantor construction, E has
log t
α
Hausdorff dimension α = log
N , and µ(B(x, r )) ≤ Cr for all
x ∈ R, r > 0.
The Fourier-analytic properties of µ depend on the choice of
subintervals.
I
If the choices of intervals are always the same (e.g. the
middle-thirds Cantor set), E and µ have arithmetic structure;
Fourier-analytic behaviour analogous to flat surfaces.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
More general Cantor measures, cont.
For any choice of subintervals in the Cantor construction, E has
log t
α
Hausdorff dimension α = log
N , and µ(B(x, r )) ≤ Cr for all
x ∈ R, r > 0.
The Fourier-analytic properties of µ depend on the choice of
subintervals.
I
If the choices of intervals are always the same (e.g. the
middle-thirds Cantor set), E and µ have arithmetic structure;
Fourier-analytic behaviour analogous to flat surfaces.
I
”Random” Cantor sets (with the subintervals chosen through
a randomized procedure) can behave like curved hypersurfaces
such as spheres.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Pointwise Fourier decay:
curvature vs. flatness,
randomness vs. structure
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Pointwise decay of µ
b: curvature and randomness
Define the Fourier transform µ
b(ξ) =
When do we have an estimate
R
e −2πiξ·x dµ(x).
|b
µ(ξ)| ≤ C (1 + |ξ|)−β/2
for some β > 0?
Izabella Laba
Harmonic analysis and the geometry of fractals
(2)
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Pointwise decay of µ
b: curvature and randomness
Define the Fourier transform µ
b(ξ) =
When do we have an estimate
R
e −2πiξ·x dµ(x).
|b
µ(ξ)| ≤ C (1 + |ξ|)−β/2
for some β > 0?
I
If µ is supported on a hyperplane {x = (x1 , . . . , xd ) ∈ Rd :
x1 = 0}, then µ
b(ξ) does not depend on ξ1 , hence no such
estimate is possible.
Izabella Laba
Harmonic analysis and the geometry of fractals
(2)
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Pointwise decay of µ
b: curvature and randomness
Define the Fourier transform µ
b(ξ) =
When do we have an estimate
R
e −2πiξ·x dµ(x).
|b
µ(ξ)| ≤ C (1 + |ξ|)−β/2
(2)
for some β > 0?
I
If µ is supported on a hyperplane {x = (x1 , . . . , xd ) ∈ Rd :
x1 = 0}, then µ
b(ξ) does not depend on ξ1 , hence no such
estimate is possible.
I
But if µ = σ is the surface measure on the sphere S d−1 , then
(2) holds with β = d − 1.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Pointwise decay of µ
b: curvature and randomness
Define the Fourier transform µ
b(ξ) =
When do we have an estimate
R
e −2πiξ·x dµ(x).
|b
µ(ξ)| ≤ C (1 + |ξ|)−β/2
(2)
for some β > 0?
I
If µ is supported on a hyperplane {x = (x1 , . . . , xd ) ∈ Rd :
x1 = 0}, then µ
b(ξ) does not depend on ξ1 , hence no such
estimate is possible.
I
But if µ = σ is the surface measure on the sphere S d−1 , then
(2) holds with β = d − 1.
I
For surface measures, estimates such as (2) depend on
curvature.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Fourier decay for fractal sets
I
Structured case: let µ be the middle-third Cantor measure,
then µ
b(3) = µ
b(32 ) = · · · = µ
b(3j ) = . . . , hence no pointwise
decay.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Fourier decay for fractal sets
I
Structured case: let µ be the middle-third Cantor measure,
then µ
b(3) = µ
b(32 ) = · · · = µ
b(3j ) = . . . , hence no pointwise
decay.
I
Salem measures: can take β arbitrarily close to dimH (supp µ).
(This is essentially the best possible decay.) Constructions due
to Salem, Kahane, Kaufman, Bluhm, Laba-Pramanik, Chen...
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Fourier decay for fractal sets
I
Structured case: let µ be the middle-third Cantor measure,
then µ
b(3) = µ
b(32 ) = · · · = µ
b(3j ) = . . . , hence no pointwise
decay.
I
Salem measures: can take β arbitrarily close to dimH (supp µ).
(This is essentially the best possible decay.) Constructions due
to Salem, Kahane, Kaufman, Bluhm, Laba-Pramanik, Chen...
I
Most constructions of Salem measures (all except Kaufman)
are probabilistic. Fourier decay is a measure of the
“randomness” of µ.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Restriction estimates
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Restriction estimates for measures on Rd
d
For f ∈ L1 (dµ), define fdµ(ξ)
=
we have an estimate
R
f (x)e −2πiξ·x dµ(x). When do
d p d ≤ Cp,q kf kLq (dµ) ?
kfdµk
L (R )
Large body of work in classical harmonic analysis (Stein, Tomas,
Fefferman, Bourgain, Tao, Wolff, Christ, Vargas, Carbery, Seeger,
Bak, Oberlin, Guth, ...). The range of exponents depends on the
geometrical properties of µ.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
L2 restriction theorem
Let µ be a compactly supported probability measure on Rd such
that for some α, β ∈ (0, d)
I
µ(B(x, r )) ≤ C1 r α for all x ∈ Rd , r > 0,
I
|b
µ(ξ)| ≤ C2 (1 + |ξ|)−β/2
Then for all p ≥ pd,α,β :=
2(2d−2α+β)
,
β
d p d ≤ Cp kf kL2 (dµ)
kfdµk
L (R )
for all f ∈ L2 (dµ).
Stein-Tomas (1970s) for the sphere, Mockenhaupt, Mitsis (2000),
Bak-Seeger (2011) general case.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Range of restriction exponents: manifolds
I
If µ = σ is the surface measure on the sphere, the
Stein-Tomas range of exponents is known to be optimal.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Range of restriction exponents: manifolds
I
If µ = σ is the surface measure on the sphere, the
Stein-Tomas range of exponents is known to be optimal.
I
Seen from Knapp example: characteristic functions of small
spherical caps of diameter δ. (The sphere is curved, but
spherical caps become almost flat as δ → 0. Equivalently, the
sphere is tangent to flat hyperplanes, with the Stein-Tomas
range of exponents reflecting the degree of tangency.)
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Range of restriction exponents: manifolds
I
If µ = σ is the surface measure on the sphere, the
Stein-Tomas range of exponents is known to be optimal.
I
Seen from Knapp example: characteristic functions of small
spherical caps of diameter δ. (The sphere is curved, but
spherical caps become almost flat as δ → 0. Equivalently, the
sphere is tangent to flat hyperplanes, with the Stein-Tomas
range of exponents reflecting the degree of tangency.)
I
Similar examples can be constructed for other manifolds. But
for fractal measures, the situation is more complicated...
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Range of restriction exponents: fractal measures
I
“Knapp example for fractals” (Hambrook-Laba 2012, further
work by Chen): Random Cantor sets can contain much smaller
subsets that are arithmetically structured. It follows that the
range of exponents in Mockenhaupt’s theorem is sharp.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Range of restriction exponents: fractal measures
I
“Knapp example for fractals” (Hambrook-Laba 2012, further
work by Chen): Random Cantor sets can contain much smaller
subsets that are arithmetically structured. It follows that the
range of exponents in Mockenhaupt’s theorem is sharp.
I
The construction draws on ideas from additive combinatorics,
especially restriction estimates for discrete sets (integers, finite
fields).
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Range of restriction exponents: fractal measures
I
“Knapp example for fractals” (Hambrook-Laba 2012, further
work by Chen): Random Cantor sets can contain much smaller
subsets that are arithmetically structured. It follows that the
range of exponents in Mockenhaupt’s theorem is sharp.
I
The construction draws on ideas from additive combinatorics,
especially restriction estimates for discrete sets (integers, finite
fields).
I
On the other hand, there are fractal measures for which
restriction estimates hold for a better range of exponents.
(Chen 2012, based on a construction of Körner.) Thus, a
wider range of behaviours than for smooth manifolds.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Restriction estimates beyond L2 ?
For manifolds, there are restriction estimates beyond q = 2. These
carry additional geometric information beyond curvature and
Fourier decay (e.g. Kakeya-type results in the case of the sphere).
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Restriction estimates beyond L2 ?
For manifolds, there are restriction estimates beyond q = 2. These
carry additional geometric information beyond curvature and
Fourier decay (e.g. Kakeya-type results in the case of the sphere).
Open question: Is there an analogue of this for Cantor sets on the
line?
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Maximal operators and
differentiation theorems
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Classic result: Hardy-Littlewood maximal theorem
Given f ∈ L1 (Rd ), define
1
Mf (x) = sup
r >0 |B(x, r )|
Z
|f (y )|dy
B(x,r )
Then kMf kp ≤ Cp,d kf kp for all 1 < p ≤ ∞.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Classic result: Hardy-Littlewood maximal theorem
Given f ∈ L1 (Rd ), define
1
Mf (x) = sup
r >0 |B(x, r )|
Z
|f (y )|dy
B(x,r )
Then kMf kp ≤ Cp,d kf kp for all 1 < p ≤ ∞.
Corollary (Lebesgue differentiation theorem): for a.e. x,
Z
1
lim
f (y )dy = f (x).
r →0 |B(x, r )| B(x,r )
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Maximal theorems for singular measures
Let µ be a probability measure on Rd , singular w.r.t. Lebesgue.
I
Define
Z
Mµ f (x) = sup
|f (x + ry )|dµ(y )
r >0
For what range of p is Mµ bounded on Lp (Rd )?
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Maximal theorems for singular measures
Let µ be a probability measure on Rd , singular w.r.t. Lebesgue.
I
Define
Z
Mµ f (x) = sup
|f (x + ry )|dµ(y )
r >0
For what range of p is Mµ bounded on Lp (Rd )?
I
Is there a differentiation theorem: for f ∈ Lp (Rd ), some range
of p,
Z
lim f (x + ry )dy = f (x), a.e. x?
r →0
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Maximal theorems for lower-dimensional manifolds
I
The spherical maximal operator
Z
Mσ f (x) = sup
|f (x + ry )|dσ(y )
r >0
Sd−1
d
. This range of
is bounded on Lp (Rd ) for d ≥ 2 and p > d−1
p is optimal. (Stein 1978 for d ≥ 3, Bourgain 1986 for d = 2)
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Maximal theorems for lower-dimensional manifolds
I
The spherical maximal operator
Z
Mσ f (x) = sup
|f (x + ry )|dσ(y )
r >0
Sd−1
d
. This range of
is bounded on Lp (Rd ) for d ≥ 2 and p > d−1
p is optimal. (Stein 1978 for d ≥ 3, Bourgain 1986 for d = 2)
I
Many other results on maximal and averaging operators
associated with smooth lower-dimensional manifolds (Stein,
Wainger, Nagel, Sogge, Phong, Iosevich, Seeger, Rubio de
Francia, ...) Most are based on curvature via the decay of µ
b.
(Notable exception: Bourgain.)
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Maximal theorems for fractals on the line
I
Laba-Pramanik 2011: For any 0 < < 13 , there is a
probability measure µ = µ supported on a set E ⊂ [1, 2] of
Hausdorff dimension 1 − , such that Mµ is bounded on Lp (R)
1+
.
for p > 1−
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Maximal theorems for fractals on the line
I
Laba-Pramanik 2011: For any 0 < < 13 , there is a
probability measure µ = µ supported on a set E ⊂ [1, 2] of
Hausdorff dimension 1 − , such that Mµ is bounded on Lp (R)
1+
.
for p > 1−
I
Implies a differentiation theorem with the same range of p.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Maximal theorems for fractals on the line
I
Laba-Pramanik 2011: For any 0 < < 13 , there is a
probability measure µ = µ supported on a set E ⊂ [1, 2] of
Hausdorff dimension 1 − , such that Mµ is bounded on Lp (R)
1+
.
for p > 1−
I
Implies a differentiation theorem with the same range of p.
I
The given ranges of and p are not likely to be optimal. We
require supp µ ⊂ [1, 2] so that µ = δ0 is not allowed.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Maximal theorems for fractals on the line
I
Laba-Pramanik 2011: For any 0 < < 13 , there is a
probability measure µ = µ supported on a set E ⊂ [1, 2] of
Hausdorff dimension 1 − , such that Mµ is bounded on Lp (R)
1+
.
for p > 1−
I
Implies a differentiation theorem with the same range of p.
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The given ranges of and p are not likely to be optimal. We
require supp µ ⊂ [1, 2] so that µ = δ0 is not allowed.
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Probabilistic construction. (Open question: deterministic
examples?) Key property of µ : a “correlation condition,”
reminiscent of higher-order uniformity conditions in additive
combinatorics.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Szemerédi-type problems
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Additive combinatorics: Szemerédi theorem
I
Szemerédi’s Theorem: Let k ≥ 3, A ⊂ {1, 2, . . . , N},
|A| ≥ δN for some δ > 0. If N is sufficiently large (depending
on δ, k), then A must contain a k-term arithmetic progression
{x, x + r , . . . , x + (k − 1)r } with r 6= 0.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Additive combinatorics: Szemerédi theorem
I
Szemerédi’s Theorem: Let k ≥ 3, A ⊂ {1, 2, . . . , N},
|A| ≥ δN for some δ > 0. If N is sufficiently large (depending
on δ, k), then A must contain a k-term arithmetic progression
{x, x + r , . . . , x + (k − 1)r } with r 6= 0.
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Brief history: Roth (1953, k = 3), Szemerédi (1969-74, all k),
Furstenberg (1977), Gowers (1998), Gowers and
Nagle-Rödl-Schacht-Skokan (2003-04); more...
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Additive combinatorics: Szemerédi theorem
I
Szemerédi’s Theorem: Let k ≥ 3, A ⊂ {1, 2, . . . , N},
|A| ≥ δN for some δ > 0. If N is sufficiently large (depending
on δ, k), then A must contain a k-term arithmetic progression
{x, x + r , . . . , x + (k − 1)r } with r 6= 0.
I
Brief history: Roth (1953, k = 3), Szemerédi (1969-74, all k),
Furstenberg (1977), Gowers (1998), Gowers and
Nagle-Rödl-Schacht-Skokan (2003-04); more...
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Many extensions and generalizations, including a
multidimensional version (Furstenberg-Katznelson 1978) and
the polynomial Szemerédi theorem (Bergelson-Leibman 1996)
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
A continuous analogue?
I
First attempt: Let A ⊂ R be a finite set, e.g.
A = {0, 1, 2, . . . , k − 1}. Must any set E ⊂ [0, 1] of positive
Lebesgue measure contain an affine (i.e. rescaled and
translated) copy of A?
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
A continuous analogue?
I
First attempt: Let A ⊂ R be a finite set, e.g.
A = {0, 1, 2, . . . , k − 1}. Must any set E ⊂ [0, 1] of positive
Lebesgue measure contain an affine (i.e. rescaled and
translated) copy of A?
I
Too easy! Positive answer follows immediately from the
Lebesgue differentiation (or density) theorem.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
A continuous analogue?
I
First attempt: Let A ⊂ R be a finite set, e.g.
A = {0, 1, 2, . . . , k − 1}. Must any set E ⊂ [0, 1] of positive
Lebesgue measure contain an affine (i.e. rescaled and
translated) copy of A?
I
Too easy! Positive answer follows immediately from the
Lebesgue differentiation (or density) theorem.
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Erdős: Same question if A is an infinite sequence. There are
counterexamples for slowly decaying sequences (Falconer,
Bourgain) but open e.g. for A = {2−n : n = 1, 2, . . . }.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
A continuous analogue: finite patterns in sets of measure
zero
Let A ⊂ R be a finite set, e.g. A = {0, 1, 2}. If a set E ⊂ [0, 1] has
Hausdorff dimension α sufficiently close to 1, must it contain an
affine copy of A?
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
A continuous analogue: finite patterns in sets of measure
zero
Let A ⊂ R be a finite set, e.g. A = {0, 1, 2}. If a set E ⊂ [0, 1] has
Hausdorff dimension α sufficiently close to 1, must it contain an
affine copy of A?
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Keleti 1998: There is a closed set E ⊂ [0, 1] of Hausdorff
dimension 1 (but Lebesgue measure 0) which contains no
affine copy of {0, 1, 2}.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
A continuous analogue: finite patterns in sets of measure
zero
Let A ⊂ R be a finite set, e.g. A = {0, 1, 2}. If a set E ⊂ [0, 1] has
Hausdorff dimension α sufficiently close to 1, must it contain an
affine copy of A?
I
Keleti 1998: There is a closed set E ⊂ [0, 1] of Hausdorff
dimension 1 (but Lebesgue measure 0) which contains no
affine copy of {0, 1, 2}.
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But additive combinatorics suggests that there should be
positive results under additional pseudorandomness conditions
on E .
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Arithmetic progressions in fractal sets
Laba-Pramanik 2009: Let E ⊂ [0, 1] compact. Assume that E
supports a probability measure µ such that:
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µ(B(x, r )) ≤ C1 r α for all x, r ∈ [0, 1] (in particular,
dim(E ) ≥ α),
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|b
µ(k)| ≤ C2 (1 + |k|)−β/2 for all k ∈ Z and some β > 2/3.
If α is close enough to 1 (depending on C1 , C2 ), then E contains a
non-trivial 3-term arithmetic progression.
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Multidimensional version
Chan-Laba-Pramanik 2013: multidimensional version. Full
statement too technical but here is an example. Let a, b, c ∈ R2
distinct. Let E ⊂ R2 compact, supports a probability measure µ
such that:
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µ(B(x, r )) ≤ C1 r α for all x ∈ R2 , r > 0,
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|b
µ(ξ)| ≤ C2 (1 + |ξ|)−β/2 for all ξ ∈ R2 .
If α, β are close enough to 2, then E must contain three distinct
points x, y , z such that 4xyz is a similar copy of 4abc.
(Maga 2012: not true without the Fourier decay assumption.)
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Ideas from additive combinatorics
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Multilinear forms “counting” arithmetic progressions, defined
via Fourier analysis.
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Decomposition of µ into “structured” and “random” parts.
The structured part is absolutely continuous and contributes
the main term to the multilinear form. The random part
contributes small errors. (Idea from Green 2003, Green-Tao
2004.)
Izabella Laba
Harmonic analysis and the geometry of fractals
Measures, randomness, Fourier decay
Restriction estimates
Maximal operators and differentiation theorems
Szemerédi-type problems
Thank you!
Izabella Laba
Harmonic analysis and the geometry of fractals